Up until this point, our focus has been on deriving the governing equations
for incompressible irrotational flows. In this lesson let's go a step further and look at how to
obtain the solutions of these equations for some simple flow fields. For all the flow fields that
we will look at in this lesson, the solution strategy will be very similar: We will first
define the flow field based on certain known para meters such as the velocity of the flow.
We will then derive the governing equations for the stream function or the velocity potential.
Further later in the course, we will see how we can superimpose multiple functions obtained from
simple flow-fields to create a complicated and realistic fluid flow problem.
The first elementary flow-field we will consider is that of a uniform flow, I.e,
constant velocities in the x and y directions. From the definition of the velocity potential,
we know that the gradient of the velocity potential is the velocity of the
fluid. Integrating these two equations with respect to x and y directions respectively,
we get two governing equations for the same variable I.e. the velocity potential. By
comparing these two relations, we can write the functions f(y) and g(x) as shown
here. Substituting for these in the velocity potential equation,
we get the following governing equation for the uniform flow. In
a similar fashion, based on the definition of the stream function, we have the following relations
that relate the velocity components to the gradients of the stream function.
I will let you know work it out yourself but after Integrating, comparing and substituting,
we obtain the governing equation for stream function for this particular flow-field
In order to visualize the profiles for these functions,
let us assign the numerical values to the two velocity components.
BY doing so, we will be able to plot the streamlines and equipotential lines
as shown here. Streamlines, as discussed previously, describe the direction of motion
of the fluid. The equipotential lines on the other hand are orthogonal to the streamlines.
Let's now look at another case.
Imagine a two dimensional flow where the fluid is emanating radially outward from a single point.
Based on this information, we can immediately draw the streamlines which look like this.
Such a flow is called a source flow and the point is referred to as the source.
If instead, we have the fluid radially flow inward towards a point, we call the point as a sink
and the flow is referred to as the sink flow. You can think of the sink flow as a negative
source flow. These flows are described in the cylindrical co-ordinate system with the three axes
being r,theta, and z, where z is perpendicular to the screen. Let us consider just the source flow.
If we now specify that the velocity of the fluid reduces as it moves radially away from the source,
we can write that the radial component of velocity in the following form.
Here, c is a proportionality constant defined as lambda/2*pi*r, where lambda is the strength of the
source. A detailed derivation is given in the supplemental material on the course page. And
since we are not specifying any rotational velocity, the angular velocity is 0.
Now that we have the velocity components, by following the same process we did for the
uniform flow problem, we can calculate the stream function and the velocity potential equations.
Assuming a numerical value for the source strength, we can plot the above equations to show
the orientation of the streamlines and the equipotential lines for the source flow. Here
is a challenge for you. Now that you know how to solve for a source flow,
try to obtain the solution for a sink flow. Here is the solution you should get.
To get your started, here is a big clue -- SINK FLOW IS JUST A NEGATIVE SOURCE FLOW.
Let us now shift gears and create a new flow based on the principle of
superposition using the uniform, source, and the sink flow that we just looked at.
As we discussed in the previous lesson, the Laplace equation is a linear equation, which means
we can combine multiple solutions of the laplace equation to create a new solution.
this is the principle of superposition.
We have already derived that the stream function is a solution of the laplace equation.
let us now create a new stream function which is the sum of the stream functions of the uniform,
source and the sink flows. the equation written in cylindrical coordinate system is shown here.
For the uniform flow, we are enforcing that the flow is strictly horizontal I.e. the v component
of velocity is 0. If we now graph this new stream function, we can notice two distinct zones.
The source and sink appear to create an elliptical envelope and the uniform flow
appears to be deflected around this envelope. As such, there is no interaction between the outside
and inside flows . As a result, it is possible to replace the flow inside the oval
with a solid body that has the same shape as that of the oval. By doing so, we have just created a
flow-field for an incompressible, irrotational, inviscid flow around an oval shaped solid body!
This is a significant milestone in fluid dynamics since it has enabled the engineers to come-up
with solutions Based on a combination of elementary Potential flows.
The flow around the Oval shaped body was first studied
by the Scottish engineer Rankine and hence is also called the Rankine oval.
It is time to introduce our third Elementary flow that is a doublet or a dipole. Here, we consider
a source and sink pair of equal but opposite strength separated by a distance, say "l".
The streamfunction for this combination can be simply written as shown here.
Let us now start reducing the distance between the source and sink. However, we will proportionally
increase the strength of the source and sink such that the product of strength
and the distance between the source and sink is constant. Let us denote this product
as kappa and call it the strength of the doublet. We can then write the stream function as shown,
which upon solving gives the following equation for the stream function of a doublet.
The plot of this stream function shows a double-lobed circular flow. You might be thinking,
so, how is this useful? Well, let us now combine this solution with a uniform horizonal flow.
The equation for the stream function we get is shown here. Graphing this equation shows
something that is very profound. Similar to what we have with the Rankine Oval,
we now have two distinct regions, that do not interact, separated by an circular envelope.
The region inside the envelope can be replaced by a circular cylinder
of the radius R, where R depends on the velocity of free stream and the strength of the doublet.
This flow-field now describes an incompressible, irrotational, inviscid flow around a cylinder.
We can get the pressure distribution on the surface of the cylinder by using the Bernoulli's
principle. If we plot the pressure distribution, we notice that the upper and lower pressure
distributions are symmetrical I.e. there is no net vertical force I.e lift force on the body.
Also, the front and the back pressure distribution is symmetrical about the vertical axis.
I.e., there is no net horizonal force or pushing force on the cylinder.
Digressing here for bit, I am sure you must have put your hand out of a moving car
and felt the air push your hand away. This force of the air is called the drag force. Even
scientists back in the 18th century must have put their hands out of a horse carriage
and felt this pushing force. Strangely, our calculations just showed that there
is no net horizonal force or drag force on the cylinder. This is called D'Alembert's Paradox,
named after the scientist who first encountered this dilemma. Here is a picture that shows how
streamlines lines look like around a cylinder in a real world case. Can you imagine that it
was a mystery for the scientists for about two centuries as to what causes this behavior? Luckily
for us, we now know that it is because of the inviscid assumption that results in zero drag
force on bodies in potential flows. Let us now consider our fourth and final
elementary flow I.e. vortex flow. Let us imagine that we have a flow
where the streamlines are concentric circles around a point, as shown in the image here.
We will impose that the velocity along each of the streamlines is constant but, it varies between
the streamlines such that the angular velocity is inversely proportional the distance
of the streamline from the center. Such a flow is called the vortex flow or a potential vortex.
For this flow, we have the following form for the angular velocity along a streamline.
In order to calculate the value of constant, let us recall the definition of circulation.
Circular is the line integral of velocity along a closed curve. Since the streamlines in the vortex
flow are closed curves, the circulation strength for each of these streamlines
in terms of the angular velocity can be written as below. Comparing with the definition of angular
velocity in the vortex flow, we immediately notice that the constant is dependent on the circulation.
Using the same process as all the previous potential flow problems we discussed in this
lesson, we can derive the governing equations for the vortex flow both in terms of velocity
potential and the stream function. Previously, we saw how a combination the uniform flow
with a source and sink flow, resulted in the flow around a moving cylinder.
If we now superpose the vortex flow with the uniform flow past a circular cylinder,
we will arrive at the solution for a flow past a rotating cylinder. It is a straightforward process
to extract the velocity components. From the velocity of the rotation of the cylinder,
we can extract the solution for the pressure co-efficient.
If we plot the pressure distribution, we obtain an asymmetric distribution between the upper
and lower surfaces of the cylinder. This indicates that there is a net force in the vertical
direction which we refer to as the lifting force. The lift force is proportional to the circulation
and the equation relating the two which we see here is the famous Kutta Joukowski theorem.
Even today, many design studies in problems that involve asymmetric pressure distribution
on a solid body, like an aircraft wing,
use this theorem to get a first estimate of the lift force.